Bifurcating vortex solutions of the complex Ginzburg-Landau equation

Hans G. Kaper; Peter Takáč

doi:10.3934/dcds.1999.5.871

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1999, Volume 5, Issue 4: 871-880. Doi: 10.3934/dcds.1999.5.871

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Bifurcating vortex solutions of the complex Ginzburg-Landau equation

Hans G. Kaper^1, and
Peter Takáč^2,

1.
Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439
2.
Fachbereich Mathematik, Universität Rostock, Universitätsplatz 1, D-18055 Rostock

Received: December 1998

Revised: June 1999

Published: October 1999

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Abstract

It is shown that the complex Ginzburg-Landau (CGL) equation on the real line admits nontrivial $2\pi$-periodic vortex solutions that have $2n$ simple zeros ("vortices") per period. The vortex solutions bifurcate from the trivial solution and inherit their zeros from the solution of the linearized equation. This result rules out the possibility that the vortices are determining nodes for vortex solutions of the CGL equation.

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Mathematics Subject Classification: 35K55, 35Q35, 58F14.

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